3 SPECTRUM OF A COMMUTATIVE BANACH ALGEBRA 331 with exponents /c^O. To each function fe <3^p(^) there corresponds the analytic function in the disk | z | < 1 defined by F(z) 2772 For 1 ^p — < oo (resp. p = -f oo) /is the limit (resp. weak limit) in ^"(ju-) of the functions 0t->F(reie) as n->l, so that Hp(/x) may be identified with the space of analytic functions F corresponding to the functions /e Jfp(/x). (Use the fact that ^P(JJL) is the closure (resp. weak closure) of B in «^(ju,).) In particular, H2(^) is identified with the space of functions which are analytic for \z\<\ and such that JT \cn\2 < +00 (cf. the problem in n = 0 Section 9.13). (b) Let v be a measure disjoint from /x which is quasi-representative for %. Show that e~t$ - v is also a quasi-representative measure for #, and hence by induction that e-nio . v is a quasi-representative measure for %, for all integers n > 0. Hence show that v = 0 and consequently (Problem 14) that every quasi-representative measure for x has base ju,. (c) Let/e 3^l(\L} be nonnegligible (with respect to fx). Then the function log|/| is ju,-integrable, or equivalently, JM(|/|) > 0, and consequently / is the product of an exterior function and an interior function. (This follows from Problem ll(b) if \fdfji 7* 0. If \fdfji = 0 then/, identified with an analytic function on the unit disk, is of the form /(z) = zng(z) where n > 0 and 0(0) ^ 0. We have g e ^'(ju.) and log |/| = log \g\ on U.) In particular, a function belonging to ^(p) which vanishes on a non-ju,-negligible subset of U is /z-negligible. Every function in ^(/u.) is the product of two functions in .^2(/x) and conversely (Problem 15(e)). A function /^> 0 belonging to •£>l(p) is equivalent to a function of the form \g \2, where g e Jf 2(ju,), if and only if JM(/) > 0 (or equivalently if and only if log |/| is p-integrable). (See Section 22.19, Problem 19.) 17. Let B be the Banach algebra considered in Problem 16, and BI the Banach subalgebra of B consisting of functions/continuous on |z| <j 1 and analytic on |z| < 1, and such that/'(0) = 0. For each complex number c such that c\ <j 1, let ftc denote the measure Fc • /x, where Fc(z) = 1 — ^(cz). Show that the measures /xc are representative meas- ures for the character ^ : /t—»/(0); the representative measures which are extremal points of 9R(x) are *ne Me suc^ that |c| — 1; the measure p,o = p, is the unique Jensen measure for xi and hence there exists no hyperextremal representative measure for x* 18. Let A be a commutative Banach algebra with unit element. (a) Let u e A be such that p(u) < 1. Then for each character ^ of A the element 1 — x(w)w is invertible. If v — (x(u) — *0(1 — x(") w)-1» tnen 1x^)1 ^ •^> (b) If xi, X* are two characters of A, let CT(XI, Xa) denote the least upper bound of the number IxaWI as u runs through the set of elements of A such that p(u) <J 1 and xi(w) = 0. Show that if p(u) £ 1 we havethe closure of B0M in ^(p), deduce that/i